On Fuzzy Matroids

The aim of this paper is to discuss properties of fuzzy regular-flats, fuzzy C- flats, fuzzy alternative-sets and fuzzy i-flats. Moreover, we characterize some peculiar fuzzy matroids via these notions. Finally, we provide a decomposition of fuzzy strong maps

Graphs and Matroids Weighted in a Bounded Incline Algebra

Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and themost reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied

LL-fuzzy ideal degrees in effect algebras

summary:In this paper, considering $L$ being a completely distributive lattice, we first introduce the concept of $L$-fuzzy ideal degrees in an effect algebra $E$, in symbol $\mathfrak{D}_{ei}$. Further, we characterize $L$-fuzzy ideal degrees by cut sets. Then it is shown that an $L$-fuzzy subset $A$ in $E$ is an $L$-fuzzy ideal if and only if $\mathfrak{D}_{ei}(A)=\top,$ which can be seen as a generalization of fuzzy ideals. Later, we discuss the relations between $L$-fuzzy ideals and cut sets ($L_{\beta}$-nested sets and $L_{\alpha}$-nested sets). Finally, we obtain that the $L$-fuzzy ideal degree is an $(L,L)$-fuzzy convexity. The morphism between two effect algebras is an $(L,L)$-fuzzy convexity-preserving mapping

Fuzzy bases and the fuzzy dimension of fuzzy vector spaces

In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space $(E,mu)$ is a fuzzy set $beta$ on $E$. The cardinality of a fuzzy basis $beta$ is called the fuzzy dimension of $(E,mu)$. The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have the same cardinality. If $widetilde{E}_1$ and $widetilde{E}_2$ are two fuzzy vector spaces, then $dim(widetilde{E}_1+widetilde{E}_2)+dim(widetilde{E}_1cap widetilde{E}_2)=dim(widetilde{E}_1) +dim(widetilde{E}_2)$ and $dim({widetilde{rm{ker }}f})+dim({widetilde{rm{im }}f})=dim(widetilde{E})$ hold without any restricted conditions. end{abstract

International Journal of Mathematical Combinatorics, Vol.6

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Fuzzy Techniques for Decision Making 2018

Zadeh's fuzzy set theory incorporates the impreciseness of data and evaluations, by imputting the degrees by which each object belongs to a set. Its success fostered theories that codify the subjectivity, uncertainty, imprecision, or roughness of the evaluations. Their rationale is to produce new flexible methodologies in order to model a variety of concrete decision problems more realistically. This Special Issue garners contributions addressing novel tools, techniques and methodologies for decision making (inclusive of both individual and group, single- or multi-criteria decision making) in the context of these theories. It contains 38 research articles that contribute to a variety of setups that combine fuzziness, hesitancy, roughness, covering sets, and linguistic approaches. Their ranges vary from fundamental or technical to applied approaches

Emergent Design

Explorations in Systems Phenomenology in Relation to Ontology, Hermeneutics and the Meta-dialectics of Design SYNOPSIS A Phenomenological Analysis of Emergent Design is performed based on the foundations of General Schemas Theory. The concept of Sign Engineering is explored in terms of Hermeneutics, Dialectics, and Ontology in order to define Emergent Systems and Metasystems Engineering based on the concept of Meta-dialectics. ABSTRACT Phenomenology, Ontology, Hermeneutics, and Dialectics will dominate our inquiry into the nature of the Emergent Design of the System and its inverse dual, the Meta-system. This is an speculative dissertation that attempts to produce a philosophical, mathematical, and theoretical view of the nature of Systems Engineering Design. Emergent System Design, i.e., the design of yet unheard of and/or hitherto non-existent Systems and Metasystems is the focus. This study is a frontal assault on the hard problem of explaining how Engineering produces new things, rather than a repetition or reordering of concepts that already exist. In this work the philosophies of E. Husserl, A. Gurwitsch, M. Heidegger, J. Derrida, G. Deleuze, A. Badiou, G. Hegel, I. Kant and other Continental Philosophers are brought to bear on different aspects of how new technological systems come into existence through the midwifery of Systems Engineering. Sign Engineering is singled out as the most important aspect of Systems Engineering. We will build on the work of Pieter Wisse and extend his theory of Sign Engineering to define Meta-dialectics in the form of Quadralectics and then Pentalectics. Along the way the various ontological levels of Being are explored in conjunction with the discovery that the Quadralectic is related to the possibility of design primarily at the Third Meta-level of Being, called Hyper Being. Design Process is dependent upon the emergent possibilities that appear in Hyper Being. Hyper Being, termed by Heidegger as Being (Being crossed-out) and termed by Derrida as Differance, also appears as the widest space within the Design Field at the third meta-level of Being and therefore provides the most leverage that is needed to produce emergent effects. Hyper Being is where possibilities appear within our worldview. Possibility is necessary for emergent events to occur. Hyper Being possibilities are extended by Wild Being propensities to allow the embodiment of new things. We discuss how this philosophical background relates to meta-methods such as the Gurevich Abstract State Machine and the Wisse Metapattern methods, as well as real-time architectural design methods as described in the Integral Software Engineering Methodology. One aim of this research is to find the foundation for extending the ISEM methodology to become a general purpose Systems Design Methodology. Our purpose is also to bring these philosophical considerations into the practical realm by examining P. Bourdieu’s ideas on the relationship between theoretical and practical reason and M. de Certeau’s ideas on practice. The relationship between design and implementation is seen in terms of the Set/Mass conceptual opposition. General Schemas Theory is used as a way of critiquing the dependence of Set based mathematics as a basis for Design. The dissertation delineates a new foundation for Systems Engineering as Emergent Engineering based on General Schemas Theory, and provides an advanced theory of Design based on the understanding of the meta-levels of Being, particularly focusing upon the relationship between Hyper Being and Wild Being in the context of Pure and Process Being